Holevo capacity achieving joint detection receiver

ABSTRACT

An optical receiver may include a unitary transformation operator to receive an n-symbol optical codeword associated with a codebook, and to perform a unitary transformation on the received optical codeword to generate a transformed optical codeword, where the unitary transformation is based on the codebook. The optical receiver may further include n optical detectors, where a particular one of the n optical detectors is to detect a particular optical symbol of the transformed optical codeword, and to determine whether the particular optical symbol corresponds to a first optical symbol or a second optical symbol. The optical receiver may also include a decoder to construct a codeword based on the determinations, and to decode the constructed codeword into a message using the codebook. The optical receiver may attain superadditive capacity, and, with an optimal code, may attain the Holevo limit to reliable communication data rates.

REFERENCE TO RELATED APPLICATION

This application claims priority to U.S. Provisional Patent ApplicationNo. 61/430,791, filed Jan. 7, 2011, the entire contents of theprovisional application being incorporated herein by reference.

BACKGROUND

The maximum amount of information that can be transmitted over aclassical noisy communication channel is known as Shannon capacity.Shannon's theorem states that, for a noisy channel, if a transmissionrate is less than the Shannon capacity, there exist error correctioncodes that allow a probability of error at a receiver to be madearbitrarily small. Thus, given the right error correction codes, thetransmission rate may approach the Shannon capacity. When a modulationalphabet of a communication channel are quantum states, a Holevo limitis an upper bound to the Shannon capacity of the communication channelcoupled with any receiver. Achieving the Holevo limit for acommunication channel may be quite challenging.

SUMMARY

According to one aspect, an optical receiver may include a unitarytransformation device to receive an n-symbol optical codeword associatedwith a codebook, and to perform an optical unitary transformation on thereceived optical codeword to generate a transformed optical codeword,where the unitary transformation is based on the codebook. The opticalreceiver may further include n optical detectors, where a particular oneof the n optical detectors is to detect a particular optical symbol ofthe transformed optical codeword, and to determine whether theparticular optical symbol corresponds to at least a first optical symbolor a second optical symbol. The optical receiver may also include adecoder to construct a codeword based on the determinations, and todecode the constructed codeword into a message using the codebook.

According to another aspect, a method, performed by an optical receiver,may include: receiving, by the optical receiver, an optical codewordassociated with a codebook, the optical codeword including n symbols;performing, by the optical receiver, a unitary transformation on thereceived optical codeword to generate a transformed optical codewordincluding n transformed symbols, which may be an entangled state of nsymbols, where the unitary transformation corresponds to a losslessrotation on the quantum states of the n symbols; n optical detectorsdetecting the n transformed symbols; determining, by the opticalreceiver and for the outputs of the n optical detectors detecting the ntransformed symbols, whether a particular one of the n transformedsymbols corresponds to at least a first optical symbol or a secondoptical symbol; and generating, by the optical receiver, an electricalestimate of the codeword, corresponding to the optical codeword, basedon the determination.

According to yet another aspect, a method, performed by a computerdevice, may include: receiving, by the computer device, a codebook;generating, by the computer device, a minimum probability of error (MPE)basis for the codebook; generating, by the computer device, ameasurement unitary matrix for the codebook by expressing codewords ofthe codebook in the MPE basis; generating, by the computer device, asingle mode MPE basis based on representations of single modemeasurements on binary symbols used to represent the codewords of thecodebook; generating, by the computer device, a single mode measurementunitary matrix for the codebook by expressing the codewords of thecodebook in the single mode MPE basis; and generating, by the computerdevice, a unitary transformation operator for the codebook bymultiplying an inverse of the measurement unitary matrix with the singlemode measurement unitary matrix.

BRIEF DESCRIPTION OF THE DRAWINGS

The accompanying drawings, which are incorporated in and constitute apart of this specification, illustrate the invention and, together withthe description, explain the invention. In the drawings,

FIG. 1 is a diagram illustrating an example system according to animplementation described herein;

FIG. 2A is a diagram illustrating a graph of photon informationefficiency as a function of the mean photon number per mode;

FIG. 2B is a diagram illustrating a graph of the relationship between abinary phase-shift keying modulation's Holevo limit and an ultimateHolevo limit;

FIG. 3 is a diagram illustrating example components of a transmitteraccording to an implementation described herein;

FIG. 4 is a diagram illustrating example components of a superadditivejoint-detection receiver according to an implementation describedherein;

FIG. 5 is a diagram illustrating example components of a Dolinarreceiver according to an implementation described herein;

FIG. 6 is a diagram illustrating example components of a first examplereceiver according to an implementation described herein;

FIG. 7 is a diagram illustrating example components of a second examplereceiver according to an implementation described herein;

FIG. 8 is a diagram illustrating example components of a third examplereceiver according to an implementation described herein;

FIG. 9 is a diagram illustrating example components of a fourth examplereceiver according to an implementation described herein;

FIG. 10 is a diagram illustrating example components of a fifth examplereceiver according to an implementation described herein;

FIG. 11 is a flow chart of an example process for performing asuperadditive measurement of a received codeword according to animplementation described herein;

FIG. 12 is a diagram of example components of a computer deviceaccording to an implementation described herein;

FIG. 13 is a flow chart of an example process for generating a unitarytransformation operator for a codebook according to an implementationdescribed herein;

FIG. 14 is a flow chart of an example process for generating a combinedunitary transformation operator according to an implementation describedherein; and

FIG. 15 is a flow chart of an example process for generating an opticalcircuit for a unitary transformation operator according to animplementation described herein.

DETAILED DESCRIPTION

The following detailed description of the invention refers to theaccompanying drawings. The same reference numbers may be used indifferent drawings to identify the same or similar elements.

An implementation described herein may relate to a joint-detectionoptical receiver (JDR) that can achieve superadditive channel capacity.Superadditivity may refer to a channel capacity higher than a channelcapacity realizable by making single symbol measurements, and may beachieved by making joint measurements over multiple symbols. A jointmeasurement may correspond to a measurement of a quantum state of acodeword block that includes multiple symbols. Joint measurements madeover multiple quantum state symbols for a communication channel may benecessary to achieve a Holevo limit associated with the communicationchannel.

An implementation described herein may relate to approaching a Holevolimit by performing a projective measurement. A projective measurementmay be implemented by performing a unitary transformation on a quantumcodeword, which may transform quantum states associated with symbols ofthe quantum codeword into superpositions of quantum states, followed byseparable projective measurements. A unitary transformation on anoptical codeword of n symbols may correspond to a losslesstransformation on the quantum state of the n symbols. In other words, aunitary transformation on an optical codeword in a vector space maycorrespond to a transformation that preserves an inner product.

An optical receiver described herein, which may approach the Holevolimit, may include a unitary transformation operator and one or moreseparate optical detectors. In one example, the separate opticaldetectors may include Dolinar receivers. In another example, theseparate optical detectors may include a different type of opticalreceiver, such as Kennedy receivers, Sasaki-Hirota receivers, homodynereceivers, heterodyne receivers, or single photon detectors.

A structure of the unitary transformation operator may be based on thecodebook used to encode the codewords received by the receiver. In anexample where the codebook corresponds to three two-symbol binary phaseshifting key (BPSK) states, the unitary transformation operator maycorrespond to a beam splitter that acts on the two symbols. In anexample where the codebook corresponds to a BPSK Hadamard code, theunitary transformation operator may correspond to a multi-stagebutterfly 50-50 asymmetric beam splitter circuit. The multi-stagebutterfly 50-50 asymmetric beam splitter circuit may be implemented asan integrated photonic circuit. The multi-stage butterfly 50-50asymmetric beam splitter circuit may be known as a Green Machine. Ingeneral, the unitary transformation operator may be determined for aparticular codebook and may be decomposed into a combination ofelementary optical operators corresponding to a beam splitter operator,a shifter operator, a squeezer operator, and a third-order Hamiltonianoperator, such as a Kerr non-linearity operator or a photon numberresolving operator. In one example, the unitary transformation operatormay be implemented as an integrated photonic circuit that includes acombination of a beam splitter, shifters, squeezers, and Kerrnonlinearity devices.

In some implementations, the unitary transformation operator may beimplemented using more than two-symbol (binary) states. For example, anarbitrary unitary operator on a n-symbol codeword (where each symbol ischosen from a M-ary modulation constellation, such as BPSK or QPSK) mayinclude: (a) beamsplitters, (b) phase shifters, (c) squeezers, and (d)Kerr non-linearities (or any other device that can provide a third-orderHamiltonian interaction). Photon number resolving (PNR) detectors, forinstance, may be used to provide a third-order Hamiltonian interaction,and may potentially be used in place of Kerr non-linearities.

The unitary transformation operator may be generated, for a codebook, bygenerating a minimum probability of error (MPE) basis for the codebook,and by representing each codeword of the codebook in terms of the MPEbasis to generate a first unitary matrix. A single measurement basis maybe generated based on a Kronecker product of single mode MPEmeasurements on the individual symbols used to generate codewords of thecodebook. Each codeword of the codebook may be represented in terms ofthe single measurement basis to generate a second unitary matrix, andthe unitary transformation operator may be generated by multiplying aninverse of the first unitary matrix with the second unitary matrix.

FIG. 1 is a diagram illustrating an example system 100 according to animplementation described herein. As shown in FIG. 1, system 100 may be asending (or transmitting) device 110, a free space optical (FSO)transmitter 120, a FSO receiver 160, and a receiving device 170.

Sending device 110 may generate a message and forward the message to FSOtransmitter 120. FSO transmitter 120 may encode the message, maygenerate a modulated light signal 150 and may transmit light signal 150across free space 140 to FSO receiver 160. Modulated light signal 150may include, for example, a spatially encoded signal that includes amultiple number of orthogonal spatial modes, with a particular one ofthe orthogonal spatial modes carrying some of the information of theencoded message. In another example, modulated light signal 150 mayinclude a temporally encoded signal. FSO receiver 160 may receivemodulated light signal 150, may decode the message, and may forward themessage to receiving device 170.

Sending device 110 and receiving device 170 may include any device withcommunication capability. For example, sending device 110 and receivingdevice 170 may include a personal computer or workstation, a serverdevice, a portable computer, a voice over Internet Protocol (VoIP)telephone device, a radiotelephone, a satellite, a base station and/oranother type of broadcast tower, a portable communication device (e.g. amobile phone, a global positioning system (GPS) device, or another typeof wireless device), a content recording device (e.g., a camera, a videocamera, etc.), a sensor, and/or any other type of electronic devicecapable of communicating with another electronic device. Sending device110 and receiving device 170 may include a same type of device ordifferent types of devices.

FSO transmitter 120 may include one or more devices that receive anelectrical message of k bits from sending device 110, encode theelectrical message into an electrical codeword of n bits using acodebook (where n≧k), transform the n-bit electrical codeword into ann-symbol optical codeword, and transmit the optical n-symbol codewordacross free space 140. FSO transmitter 120 may include, among othercomponents, a wave plate, a beamsplitter, an electro-optic modulator, alaser transmitter, and/or an optical waveform generator.

FSO receiver 160 may include one or more devices that receive then-symbol optical codeword from FSO transmitter 120, perform a unitarytransformation on the received n-symbol optical codeword to generate atransformed n-symbol optical codeword, measure individual symbols of thetransformed n-symbol optical codeword using n optical detectors toconvert the individual symbols into n electrical symbols, decode theconverted electrical symbols into an estimate of the k-bit message, andforward the decoded message to receiving device 170. FSO receiver 160may include, among other components, a phase-locked loop, a fibermodulator, a digital oscilloscope, and/or an optical receiver telescope.

Although FIG. 1 shows example components of system 100, in otherimplementations, system 100 may include fewer components, differentcomponents, differently arranged components, and/or additionalcomponents than depicted in FIG. 1. Additionally or alternatively, oneor more components of system 100 may perform one or more tasks describedas being performed by one or more other components of system 100.

FIG. 2A is a diagram illustrating a graph 200 of photon informationefficiency. As shown in FIG. 2A, graph 200 may illustrate a relationshipbetween bits per photon and mean photon number per mode, represented ascurve 210. Curve 210 may represent the information carrying capacity ofphotons, and may depict that no fundamental upper limit exists on theinformation that may be encoded in a photon. If one desires to encodemore information in a photon, one may need to lower the mean photonnumber per mode. Point 220 may illustrate an example of the informationcarrying capacity of photons at 10 bits per photon. According to animplementation described herein, using FSO transmitter 120 at 1.55micrometers (μm) and operating at a 1 gigahertz (GHz) modulationbandwidth, the laws of physics may permit reliable communication at 0.27Gigabits per second (Gbps) using only 3.4 picowatts (pW) of averagereceived optical power. At such low optical power levels, communicationsmay be secure, as it may be difficult for an intercepting party todetect the signal.

FIG. 2B is a diagram illustrating a graph 250 of the relationshipbetween a binary phase shift keying (BPSK) Holevo limit and an ultimateHolevo limit.

A BPSK modulation optical communication system may encode data bysending either an optical pulse or a 180° phase reversed optical pulseover each use of a physical channel. The maximum information that may besent over the channel using any binary modulation scheme, such as BPSK,is 1 bit per channel use, which may be achieved if the two opticalstates are perfectly distinguishable at the receiver. The lower a meanphoton number in each of the two BPSK states, the harder it is to tellthe two optical states apart. The quantum limit to the minimumprobability of error (MPE) of discriminating between the two opticalpulses may be expressed as

${P_{m\; i\; n} = {\frac{1}{2}\left\lbrack {1 - \sqrt{1 - ^{{- 4}N}}} \right\rbrack}},$

where N corresponds to the mean photon number in each of the two BPSKstates. The quantum limit of the MPE may be lower than any probabilityof error achievable by an optical receiver (e.g., a direct opticaldetector, a homodyne optical detector, or a heterodyne opticaldetector). The quantum limit of the MPE may be achieved using a Dolinarreceiver, which uses a single photon detector coupled with nearinstantaneous electrical to optical feedback from the electrical opticaloutput of the single photon detector to optical input of the singlephoton detector.

Curve 260 may illustrate a highest Shannon capacity achievable by theBPSK modulation system using a single symbol receiver, which may beattained with a Dolinar receiver. Thus, a maximum capacity reachable bya single symbol receiver may correspond to 2/ln 2, or 2.89 bits perphoton. Curve 270 may illustrate an ultimate Holevo limit of a jointdetection receiver that performs a joint measurement over a codewordblock of symbols using an optimal codebook. Curve 280 may illustrate theHolevo limit of a joint detection receiver that uses binary modulation.Thus, FIG. 2B may demonstrate that binary modulation (e.g., using BPSK)may be close to optimal to approaching the Holevo limit at low photonnumbers per mode.

FIG. 3 is a diagram illustrating example components of FSO 120transmitter. As shown in FIG. 3, FSO transmitter 120 may include anencoder 320 and a modulator 330.

Sending device 110 may break up a message to be sent to receiving device170 into k-bit messages. Encoder 320 may receive a k-bit message 310 andencode k-bit message 310 into an n-bit code word. The k-bit message maycorrespond to an element from the set {1, . . . 2^(nR)}, where Rcorresponds to the transmission rate in bits per channel use, or bitsper modulation symbol, n corresponds to the size of a codeword, and k=nRcorresponds to the size of message 310. Thus, message 310 may correspondto any sequence of k bits. Encoder 320 may include a 1-1 map that maps aparticular sequence of k bits to a particular codeword of n bits. Thus,each member of the set {1, . . . 2^(nR)} may be associated with aparticular n-bit codeword. The set of n-bit codewords may correspond tothe codebook.

Modulator 330 may receive a codeword from encoder 320 and may generatemodulated light signal 150 that includes optical codeword state 340.Modulator 330 may use BPSK to convert bits of the received n-bitcodeword into one of two possible light symbols |+a

and |−a

. For example, |+a

may correspond to a first phase of a photon and |−a

may correspond to a second phase of a photon, where the second phasediffers from the first phase by 180°. Modulator 330 may be implemented,for example, as a ferroelectric liquid crystal (FLC) array incorporatedinto an integrated circuit, in combination with a light source. An FLCmay use birefringence to convert a voltage polarity into a rotation ofthe optical axis of light emitted by the light source. In anotherimplementation, modulator 330 may be implemented using a different typeof device.

Although FIG. 3 shows example components of FSO transmitter 120, inother implementations, FSO transmitter 120 may include fewer components,different components, differently arranged components, and/or additionalcomponents than depicted in FIG. 3. Additionally or alternatively, oneor more components of FSO transmitter 120 may perform one or more tasksdescribed as being performed by one or more other components of FSOtransmitter 120.

FIG. 4 is a diagram illustrating example components of FSO receiver 160.As shown in FIG. 4, FSO receiver 160 may include a unitarytransformation device 410, one or more Dolinar receivers 420-A to 420-N(referred to herein collectively as “Dolinar receivers 420” andindividually as “Dolinar receiver 420”), and a decoder 430.

Unitary transformation device 410 may receive optical codeword 340 andmay perform a unitary transformation in the optical domain on thereceived optical codeword 340. The unitary transformation may correspondto a joint operation performed on all the symbols of the receivedoptical codeword 340 simultaneously. Unitary transformation device 410may output a transformed (and possibly entangled) optical quantum stateof the received optical codeword 340. While the received opticalcodeword 340 may correspond to a set of pure coherent-state BPSK symbols|+a

and |−a

(e.g., optical states corresponding to two different phases), thetransformed optical codeword may correspond to states which may not bepure BPSK symbols |+a

and |−a

, but rather may correspond to linear combinations of |+a

and |−a

(e.g., superpositions of |+a

and |−a

). Unitary transformation device 410 may forward individual symbolsassociated with the transformed optical codeword to particular ones ofDolinar receivers 420.

Dolinar receiver 420 may convert the received optical symbol into anelectrical signal corresponding to an estimate of whether the receivedoptical symbol, of the transformed optical codeword, corresponds to a|+a

symbol or a |−a

symbol. Dolinar receiver 420 is described below in more detail withreference to FIG. 5.

Decoder 430 may receive electrical signals from Dolinar receivers 420and may combine the received electrical signals into an estimate of ann-bit particular codeword of the codebook. Decoder 430 may then use a1-1 map to generate an estimate of k-bit message 310 corresponding tothe n-bit estimated codeword.

Although FIG. 4 shows example components of FSO receiver 160, in otherimplementations, FSO receiver 160 may include fewer components,different components, differently arranged components, and/or additionalcomponents than depicted in FIG. 4. Additionally or alternatively, oneor more components of FSO receiver 160 may perform one or more tasksdescribed as being performed by one or more other components of FSOreceiver 160.

FIG. 5 is a diagram illustrating example components of Dolinar receiver420. Dolinar receiver 420 may correspond to an optimum quantummechanical receiver for detecting one of two possible binary coherentstates. Dolinar receiver 420 may add a time-varying local field to anincoming signal field and may perform photon counting on the combinedfield. The local field may be controlled by a feedback loop that adjustsan amplitude and a phase of the local field based on the observed photoncounts. Dolinar receiver 420 may select one of two hypothesis H₀ and H₁,where selecting H₀ may correspond to detecting |−a

, and where selecting H₁ may correspond to detecting |+a

. For example, if Dolinar receiver 420 counts x photons within aparticular time period, and x is 0 or an even number, Dolinar receiver420 may select H₁, and if x is an odd number, Dolinar receiver 420 mayselect H₀.

As shown in FIG. 5, Dolinar receiver 420 may include a first beamsplitter 510, a feedback amplitude modulator 520, a feedback phasemodulator 530, a second beam splitter 540, a photon detector 550, aprocessor 560, and a feedback controller 570.

First beam splitter 510 may split an incoming light beam into a firstbeam 512 and a second beam 514. Second beam 514 may correspond to alocal field controlled by feedback controller 570. Feedback amplitudemodulator 520 may control an amplitude associated with second beam 514based on signals received from feedback controller 570. Feedback phasemodulator 530 may control a phase associated with second beam 514 basedon signals received from feedback controller 570.

Second beam splitter 540 may combine first beam 512 and locallymodulated second beam 514. Photon detector 550 may receive the combinedbeam from second beam splitter 540 and may perform a photon count on thereceived beam. For example, photon detector 550 may determine a numberof photons detected within a particular time period.

Processor 560 may receive a photon count from photon detector 550 andmay select one of two hypothesis H₀ and H₁, where selecting H₀ maycorrespond to detecting |−a

, and where selecting H₁ may correspond to detecting |+a

, based on the observed photon count. Feedback controller 570 maycontrol feedback amplitude modulator 520 and feedback phase modulator530 based on the observed photon count.

Although FIG. 5 shows example components of Dolinar receiver 420, inother implementations, Dolinar receiver 420 may include fewercomponents, different components, differently arranged components,and/or additional components than depicted in FIG. 5. Additionally oralternatively, one or more components of Dolinar receiver 420 mayperform one or more tasks described as being performed by one or moreother components of Dolinar receiver 420. In alternativeimplementations, a receiver other than a Dolinar receiver may be used.For example, in the context of Reed Muller codes, a Kennedy receiver maybe used instead of a Dolinar receiver.

In other example implementations, FSO receiver 160 may use differenttype of receivers. For example, FSO receiver 160 may use Kennedyreceivers. A Kennedy receiver may add a local field to a received signalfield without the use of a feedback loop. Rather, in a Kennedy receiver,the amplitude and phase of the local field may be set to correspond toH₀.

FIG. 6 is a diagram illustrating example components of a first examplereceiver 600 capable of being used by FSO receiver 160. As shown in FIG.6, receiver 600 may include a unitary transformation operator 610 and aprojective measurement receiver 620.

Unitary transformation operator 610 may correspond to a unitarytransformation on a codebook that includes three codewords 601, 602, and603. Codeword 601 may correspond to |ψ₁

=|α

|a

, codeword 602 may correspond to |ψ₂

=|α

−α

, and codeword 603 may correspond to |ψ₃

=|−α

|−α

. Unitary transformation operator 610 may include a beam splitter 615.

Beam splitter 615 may include a device with two input modes and twooutput modes and may separate an incident beam of light into a reflectedbeam and a transmitted beam. Beam splitter 615 may be implemented as tworight-angle prisms, cemented together at their hypotenuse, with thehypotenuse surface being coated with a metallic or dielectric layer.

Beam splitter 615 may take as input two quantum symbols and may outputtwo quantum symbols that are a superposition of the two input quantumsymbols. Thus, beam splitter 615 may perform a unitary transformation onan incoming codeword to generate a transformed (and possibly entangled)codeword. For example, beam splitter 615 may transform first codeword601 to first transformed codeword 611 corresponding to |φ₁

=|√{square root over (2)}α

|0

, may transform second codeword 602 to second transformed codeword 612corresponding to |φ₂

=|0

|√{square root over (2)}α

, and may transform third codeword 603 to third transformed codeword 613corresponding to |φ₃

=|0

|−√{square root over (2)}α

.

Projective measurement receiver 620 may include a single photon detector630 and a Dolinar receiver 640. Projective measurement receiver 620 mayperform a measurement on a transformed codeword and may determine towhich codeword the transformed codeword corresponds.

Receiver 600 may achieve superadditive capacity gain in comparison tousing a single Dolinar receiver (e.g., using Dolinar receiver 640 byitself). For example, using Dolinar receiver 640 by itself may result ina maximum photon efficiency of 2.8854 bits per photon, while receiver600 may result in a maximum photon efficiency of 2.9572 bits per photon.

Although FIG. 6 shows example components of receiver 600, in otherimplementations, receiver 600 may include fewer components, differentcomponents, differently arranged components, and/or additionalcomponents than depicted in FIG. 6. Additionally or alternatively, oneor more components of receiver 600 may perform one or more tasksdescribed as being performed by one or more other components of receiver600.

FIG. 7 is a diagram illustrating example components of a second receiver700 capable of being used by FSO receiver 160. As shown in FIG. 7,receiver 700 may include a unitary transformation operator 710 and aprojective measurement receiver 730.

Unitary transformation operator 710 may perform a unitary transformationon codewords based on a Hadamard matrix 701. The top row of Hadamardmatrix 701 may correspond to an ancilla mode (e.g., a locally encodedsignal that may facilitate the identification of errors in a receivedcodeword). The use of an ancilla mode as part of a transmitted codewordmay prevent receiver 700 from having to perform active-phase trackingand may improve performance of receiver 700.

Unitary transformation operator 710 may include a group of beamsplitters 711-722. If the input of unitary transformation operator 710includes a codeword with n optical symbols, then unitary transformationoperator 710 may include n*log₂(n/2) beam splitters. Each of beamsplitters 711-722 may take as input two quantum symbols of a codewordand may output two different quantum symbols.

Projective measurement receiver 730 may perform projective measurementson particular symbols of the transformed optical codeword. Theprojective measurements may include multiple single photon detectors. Aparticular single photon detector may perform photon counts for aparticular output of unitary transformation operator 710.

Given Hadamard matrix 701, unitary transformation operator 710 may, incombination with projective measurement receiver 730, output pulseposition modulation matrix 702. Thus, unitary transformation operator701 may perform a unitary operation that corresponds to a Walshtransform.

As an example, using receiver 700 with a 2048 input by 2048 output thatincludes 11,264 beam splitters (2048*log₂(2048)/2=11,264), together with2048 single photon detectors, may attain a capacity of 10 bits perphoton using BPSK modulation at a mean photon number per BPSK symbol ofonly 10⁻⁴.

Although FIG. 7 shows example components of receiver 700, in otherimplementations, receiver 700 may include fewer components, differentcomponents, differently arranged components, and/or additionalcomponents than depicted in FIG. 7. Additionally or alternatively, oneor more components of receiver 700 may perform one or more tasksdescribed as being performed by one or more other components of receiver700.

FIG. 8 is a diagram illustrating example components of a third receiver800 capable of being used by FSO receiver 160. Dolinar receiver 420 maybe implemented as a concatenation of a single mode unitary operation anda photon number resolving (PNR) detector. Single mode unitary operationscorresponding to multiple Dolinar receivers 420 may be subsumed into acombined unitary operation acting on a codebook. Thus, in anotherimplementation, FSO receiver 160, which includes a unitarytransformation operator in combination with a group of Dolinarreceivers, may be implemented as a combined unitary operation on areceived codeword, followed by detection with an array of PNR detectors.Receiver 800 may correspond to a device that includes a combined unitarytransformation operator. As shown in FIG. 8, receiver 800 may include acombined unitary transformation device 810, one or more PNR detectors820-A to 820-Q (referred to herein collectively as “PNR detectors 820”and individually as “PNR detectors 820”), and a decoder 830.

Combined unitary transformation device 810 may perform a combinedunitary transformation on optical codeword 340, where the combinedunitary transformation corresponds to a unitary transformation performedby unitary transformation device 410 followed by a series of single modeunitary transformations on particular symbols of a transformed opticalcodeword.

PNR detectors 820 may detect photons corresponding to particular symbolsof the transformed optical codeword and may perform a photon countassociated with a particular symbol. For example, PNR detector 820 maycount a number of photons detected for the particular symbol and maydetermine whether the particular symbol corresponds to BPSK symbol |+a

or BPSK symbol |−a

based on a number of photons detected. Decoder 830 may correspond todecoder 430 of FIG. 4.

Although FIG. 8 shows example components of receiver 800, in otherimplementations, receiver 800 may include fewer components, differentcomponents, differently arranged components, and/or additionalcomponents than depicted in FIG. 8. Additionally or alternatively, oneor more components of receiver 800 may perform one or more tasksdescribed as being performed by one or more other components of receiver800.

FIG. 9 is a diagram illustrating example components of a fourth receiver900 capable of being used by FSO receiver 160. While implementationsdescribed herein correspond to receivers using spatial encoding (e.g.,an n number of orthogonal spatial modes with n symbols transmittedsimultaneously), receiver 900 may correspond to an implementation usingtemporal coding (e.g., one spatial mode with one symbol transmitted at atime). Thus, while receiver 400 (or receiver 800) may receive ann-symbol codeword in one pulse, receiver 900 may receive an n-symbolcodeword in n pulses.

As shown in FIG. 9, receiver 900 may include an optical demultiplexer910, one or more optical buffers 920-A to 920-N (referred to hereincollectively as “optical buffers 920” and individually as “opticalbuffer 920”), unitary transformation device 410, one or more Dolinarreceivers 420-A to 420-N, and decoder 430.

Optical demultiplexer 910 may receive a temporally encoded codeword 901one symbol at a time and may select optical buffers 920 in sequence.Thus, optical demultiplexer 910 may send a first symbol to opticalbuffer 920-A, a second symbol to optical buffer 920-B, and so on untiloptical buffer 920-N receives the n-th symbol.

Optical buffers 920 may temporarily store particular symbols oftemporally encoded codeword 901. In one example, optical buffers 920 mayinclude optical paths (e.g., optical fibers) of different lengths sothat when the n-th symbol is received by optical buffer 920-N, all nsymbols are forwarded to unitary transformation device 410 atsubstantially the same time. For example, optical buffer 920-A mayinclude an optical path that is longer than the optical path of opticalbuffer 920-B, the optical path of optical buffer 920-B may include anoptical path that is longer than the optical path of optical buffer920-C, and so on, with optical buffer 920-N having a shortest opticalpath. In another example, optical buffers 920 may include slow lightdevices configured so that all n optical symbols exit optical buffers920 at substantially the same time.

Unitary transformation device 410, Dolinar receivers 420, and decoder430 may correspond to unitary transformation device 410, Dolinarreceivers 420, and decoder 430 of FIG. 4, respectively.

Although FIG. 9 shows example components of receiver 900, in otherimplementations, receiver 900 may include fewer components, differentcomponents, differently arranged components, and/or additionalcomponents than depicted in FIG. 9. Additionally or alternatively, oneor more components of receiver 900 may perform one or more tasksdescribed as being performed by one or more other components of receiver900.

FIG. 10 is a diagram illustrating example components of a fifth receiver1000 capable of being used by FSO receiver 160. Receiver 1000 may detectcodewords encoded using a first-order Reed-Muller code R(1,m). AReed-Muller code R(r,m) may correspond to a [n, k, d] linear code(0≦r≦m), where n=2^(m) is the length of each codeword, with K=2^(k)codewords in the code, and with a minimum distance, in the code space,of d=2^(m-r) between any two codewords.

First order Reed-Muller code R(1,m) may be a [2^(m), m+1, 2^(m-1)] codethat corresponds to a [2^(m), m, 2^(m-1)] Hadamard code, including anancilla, with the same codewords with flipped bits. An R(1,m) code,similar to a Hadamard code, may be decoded in the optical domain using ajoint detection receiver with n log₂ n/2 beamsplitters (where n=2^(m)),followed by an array of n single photon detectors, followed by a singleDolinar receiver.

As shown in FIG. 10, receiver 1000 may include a unitary transformationoperator 1010, a projective measurement receiver 1020, a switch 1030,and a Dolinar receiver 1040. Unitary transformation operator 1010 mayperform a unitary transformation on codewords, analogous to unitarytransformation operator 710 of FIG. 7. Projective measurement receiver1020 may include a group of single photon detectors that perform photoncounts for a particular output of unitary transformation operator 1010,analogous to projective measurement receiver 730. Dolinar receiver 1040may include a Dolinar receiver analogous to Dolinar receiver 640.Dolinar receiver 1040 may be switchable, via switch 1030, to any of thesingle photon detectors of projective measurement receiver 1020.

When a codeword is received by receiver 1000, a particular one of thesingle photon detectors, of projective measurement receiver 1020, mayclick (e.g., detect a photon). The particular single photon detectorthat clicks may trigger Dolinar receiver 1040. If none of the singlephoton detectors clicks, receiver 1000 may yield an erasure outcome,indicating that receiver 1000 was not able to determine which particularcodeword was received.

Receiver 1000, which may decode codewords based on a first order BPSKReed-Muller codebook, may outperform a detector based on directdetection of on-off keying. Furthermore, receiver 1000 may exhibitsuperadditive capacity for a BPSK alphabet, with a higher capacity thana codebook in the Hadamard codebook family.

Although FIG. 10 shows example components of receiver 1000, in otherimplementations, receiver 1000 may include fewer components, differentcomponents, differently arranged components, and/or additionalcomponents than depicted in FIG. 10. Additionally or alternatively, oneor more components of receiver 1000 may perform one or more tasksdescribed as being performed by one or more other components of receiver1000.

FIG. 11 is a flow chart of an example process for performing asuperadditive measurement of a received codeword according to animplementation described herein. In one implementation, the process ofFIG. 11 may be performed by FSO receiver 160. In other implementations,some or all of the process of FIG. 11 may be performed by another deviceor a group of devices separate and/or possibly remote from or includingFSO receiver 160.

The process of FIG. 11 may include receiving an optical codeword (block1110). For example, FSO receiver 160 may received an optical codewordbased on modulated light signal 150. The codeword may be received byunitary transformation device 410.

A unitary transformation may be performed on the received opticalcodeword (block 1120). For example, unitary transformation device 410may perform a unitary transformation on the received optical codeword.Unitary transformation device 410 may output a transformed (and possiblyentangled) optical quantum state of the received optical codeword. Whilethe received optical codeword may correspond to a set of pure BPSKsymbols |+a

and |−a

(e.g., photon states corresponding to two different phases), thetransformed optical codeword may correspond to states which may not bepure BPSK symbols |+a

and |−a

, but may rather correspond to linear combinations of |+a

and |−a

(e.g., superpositions of |+a

and |−a

).

Individual symbols of the transformed optical codeword may be detectedby individual receivers (block 1130). For example, unitarytransformation device 410 may forward individual symbols associated withthe transformed optical codeword to particular ones of Dolinar receivers420. Each of Dolinar receivers 420 may detect a particular transformedsymbol (e.g., corresponding to a particular linear combination of |+a

and |−a

).

The individual optical symbols may be converted into electrical signalsto generate an electrical codeword (block 1140). For example, aparticular Dolinar receiver 420 may convert the detected optical symbolinto an electrical signal corresponding to an estimate of whether thereceived optical symbol corresponds to a |+a

symbol or a |−a

symbol.

The electrical codeword may be decoded to retrieve a message (block1150). For example, decoder 430 may receive the electrical signals fromDolinar receivers 420 and may combine the received electrical signalsinto an estimate of an n-symbol particular codeword of the codebook.Decoder 430 may then use a 1-1 map to generate an estimate of a k-bitmessage corresponding to the n-symbol estimated codeword.

FIG. 12 is a diagram illustrating example components of a computerdevice 1200 that may be used to generate a unitary transformation matrixcorresponding to unitary transformation device 410, that may be used togenerate a combined unitary transformation matrix corresponding tocombined unitary transformation device 810, or that may be used togenerate a layout for a photonic circuit based on a particular unitarytransformation matrix. As shown in FIG. 12, computer device 1200 mayinclude a bus 1210, a processor 1220, a memory 1230, an input device1240, an output device 1250, and a communication interface 1260.

Bus 1210 may include a path that permits communication among thecomponents of computer device 1200. Processor 1220 may include one ormore processors, microprocessors, or processing logic (e.g., applicationspecific integrated circuits (ASICs) or field programmable gate arrays(FPGAs)) that may interpret and execute instructions. Memory 1230 mayinclude a random access memory (RAM) device or another type of dynamicstorage device that may store information and instructions for executionby processor 1220, a read only memory (ROM) device or another type ofstatic storage device that may store static information and instructionsfor use by processor 1220, a magnetic and/or optical recording memorydevice and its corresponding drive, and/or a removable form of memory,such as a flash memory.

Input device 1240 may include a mechanism that permits an operator toinput information to computer device 1200, such as a keypad, a button, apen, a touch screen, voice recognition and/or biometric mechanisms, etc.Output device 1250 may include a mechanism that outputs information tothe operator, including one or more light indicators, a display, aspeaker, etc.

Communication interface 1260 may include any transceiver-like mechanismthat enables computer device 1200 to communicate with other devicesand/or systems. For example, communication interface 1260 may include amodem, a network interface card, and/or a wireless interface card.

As described herein, computer device 1200 may perform certainoperations. Computer device 1200 may perform these operations inresponse to processor 1220 executing software instructions contained ina computer-readable medium, such as memory 1230. A computer-readablemedium may be defined as a non-transitory memory device. A memory devicemay include space within a single physical memory device or spreadacross multiple physical memory devices.

The software instructions may be read into memory 1230 from anothercomputer-readable medium, or from another device via communicationinterface 1260. The software instructions contained in memory 1230 maycause processor 1220 to perform processes that will be described later.Alternatively, hardwired circuitry may be used in place of or incombination with software instructions to implement processes describedherein. Thus, implementations described herein are not limited to anyspecific combination of hardware circuitry and software.

Although FIG. 12 shows example components of computer device 1200, inother implementations, computer device 1200 may contain fewercomponents, different components, additional components, or differentlyarranged components than depicted in FIG. 12. Additionally oralternatively, one or more components of computer device 1200 mayperform one or more tasks described as being performed by one or moreother components of computer device 1200.

FIG. 13 is a flow chart of an example process for generating a unitarytransformation operator for a codebook according to an implementationdescribed herein. In one implementation, the process of FIG. 13 may beperformed by computer device 1200. In other implementations, some or allof the process of FIG. 13 may be performed by another device or a groupof devices separate and/or possibly remote from or computer device 1200.

The process of FIG. 13 may include selecting a codebook with 2^(nR)codewords, each codeword being a sequence of n BPSK symbols |+a

and |−a

(block 1310). For example, computer device 1200 may select a particularcodebook C of n BPSK symbols. Each codeword may include a particularsequence of n BPSK symbols |+a

and |−a

, with |+a

and |−a

corresponding to two different binary states. For example, |+a

may correspond to a first phase of a photon and |−a

may correspond to a second phase of a photon, where the second phasediffers from the first phase by 180°. The number of codewords maycorrespond to K=2^(nR), where n corresponds to the number of symbols percodeword and where R corresponds to the transmission rate in bits perBPSK symbol.

An MPE basis for the codebook may be generated (block 1320). Forexample, computer device 1100 may calculate an MPE measurement oncodebook C in terms of a set of projection operators W_(C)={|w₁

, |w₂

, . . . |w_(k)

}. W_(C) may form a complete ortho-normal (CON) basis of a vector spacespanned by the codewords in C.

The codewords of codebook C may be extended into a set of all 2^(n)binary sequences of |+a

and |−a

(block 1330). For example, computer device 1200 may generate a setC^(E)={|c₁

, . . . , |c_(k)

, |c_(k+1)

, . . . , |c_(q)

}, where q=2^(n). For example, if n=3, C^(E)={{|−a

, |−a

, |−a

}, {|−a

, |−a

, |+a

}, {|−a

, |+a

, |−a

}, {|−a

, |+a

, |+a

}, {|+a

, |−a

, |−a

}, {|+a

, |−a

, |+a

}, {|+a

, |+a

, |−a

}, {|+a

, |+a

, |+a

}}.

The MPE basis may be extended to span the whole space of the 2^(n)binary sequences (block 1340). For example, computer device 1200 mayextend basis W_(c) to span the whole vector space spanned by C^(E) usingGram-Schmidt ortho-normalization. Extended basis W_(C) may berepresented as basis W={|w₁

, . . . |w_(k)

, |w_(k+1)

. . . |w_(q)

}.

Single mode MPE measurements on |+a

and |−a

may be represented as |+

and |−

(block 1350). For example, computer device 1200 may represent a singlemode MPE measurement on |+a

and |−a

, which may correspond to a detection by a Dolinar receiver or aSasaki-Hirota receiver, in terms of two projectors on the span of thetwo BPSK states |+a

and |−a

. The two projectors, |+

and |−

, may each correspond to a linear combination of the two pure states |+a

and |−a

. In other words, |+

=a₁|+a

+a₂−a

, and |−

=b₁|+a

+b₂|−a

.

A single mode measurement basis may be generated based on Kroneckerproducts of |+

and |−

(block 1360). For example, computer device 1200 may construct a completeortho-normal basis M of the vector space spanned by C^(E) using aKronecker product of |+

and |−

. In other words, M={|m₁

, |m₂

, . . . |m_(q)

}, where |m₁

=|+

|+

. . . |+

|+

, |m₂

=|+

|+

. . . |+

|−

, |m₃

=|+

|+

. . . |−

|+

, |m₄

=|+

|+

. . . |−

|−

, . . . , |m_(q)

=|−

|−

. . . |−

|−

.

Each codeword of codebook C may be represented in the extended MPE basisto generate an extended MPE unitary matrix (block 1370). For example,each code word |c_(k)

in C^(E) may be expressed in basis W as

${{\left| c_{k} \right.\rangle} = {\sum\limits_{j = 1}^{q}{{\langle\left. w_{j} \middle| c_{k} \right.\rangle}{w_{j}\rangle}}}},$

and the extended MPE unitary matrix U_(W) may be generated based on theset of coefficients, where

w_(j)|c_(k)

corresponds to the (j,k)^(th) coefficient of U_(W).

Each codeword of codebook C may be represented in the single modemeasurement basis to generate a single measurement MPE unitary matrix(block 1380). For example, each code word |c_(k)

in C^(E) may be expressed in basis M as

${c_{k}\rangle} = {\sum\limits_{j = 1}^{q}{\langle{{m_{j}{c_{k}\rangle}{m_{j}\rangle}},}}}$

and the single mode measurement MPE unitary matrix U_(M) may begenerated based on the set of coefficients, where

m_(j)|c_(k)

corresponds to the (j,k)^(th) coefficient of U_(M).

A unitary transformation operator may be generated by multiplying aninverse of the extended MPE basis unitary matrix with the single modemeasurement unitary matrix (block 1290). For example, computer device1200 may generate unitary transformation operator

${U = {\sum\limits_{k = 1}^{q}{\sum\limits_{j = 1}^{q}{u_{jk}{m_{j}\rangle}{\langle m_{k}}}}}},$

where u_(jk)=(U_(w) ⁻¹U_(M))_(jk) (represented in the M basis).

FIG. 14 is a flow chart of an example process for generating combinedunitary transformation device 810 according to an implementationdescribed herein. In one implementation, the process of FIG. 14 may beperformed by computer device 1200. In other implementations, some or allof the process of FIG. 14 may be performed by another device or a groupof devices separate and/or possibly remote from or computer device 1100.

The process of FIG. 14 may include determining a unitary transformationoperator for a particular codebook (block 1410). For example, computerdevice 1200 may determine a unitary transformation matrix correspondingto unitary transformation device 410.

A Dolinar receiver may be represented as a concatenation of asingle-mode unitary operator and a photon number resolving (PNR)detector (block 1420). For example, computer device 1200 may representDolinar receiver 420 as a combination of a single-mode unitary operatorand a PNR detector. Computer device 1200 may generate a single modeunitary operator, in a same format as the unitary transformation matrix(e.g., in a same basis), for each of the Dolinar receivers 420-A through420-N (e.g., may generate n single mode unitary operators).

The n single mode unitary operators may be combined with the unitarytransformation operator to generate a combined unitary transformationoperator (block 1430). For example, computer device 1200 may generate amatrix corresponding to the n single mode unitary operators and maygenerate a product of the unitary transformation matrix and the matrixcorresponding to the n single mode unitary operators to generate acombined unitary transformation matrix (which corresponds to thecombined unitary transformation operator).

A layout for a photonic circuit for implementing a joint-detectionreceiver based on the combined unitary transformation operator and a setof N photon counting detectors may be generated (block 1440). Forexample, computer device 1200 may generate a layout for a photoniccircuit that corresponds to combined unitary transformation device 810and n PNR detectors 820-A through 820-N.

FIG. 15 is a flow chart of an example process for generating an opticalcircuit for a unitary transformation operator according to animplementation described herein. In one implementation, the process ofFIG. 15 may be performed by computer device 1200. In otherimplementations, some or all of the process of FIG. 15 may be performedby another device or a group of devices separate and/or possibly remotefrom or computer device 1200.

The process of FIG. 15 may include determining a unitary transformationoperator for a particular codebook (block 1510). For example, computerdevice 1200 may determine a unitary transformation matrix that has beengenerated for codebook C using the process of FIG. 13.

The unitary transformation operator may be decomposed into a set ofelementary unitary operators representing a beam splitter, a phaseshifter, a squeezer, and a Kerr nonlinearity device (block 1520). Anyoptical unitary transformation on multiple optical modes may beaccomplished using a combination of beam splitters, a phase shifter, asqueezer, and any third-order Hamiltonian operator, such as a devicethat includes a Kerr nonlinearity device. In another implementation, athird-order Hamiltonian operator may be implemented by aphoton-number-resolving (PNR) detector.

A phase shifter may include a device, with one optical input and oneoptical output, which may introduce a variable phase delay into theoptical input. In one implementation, a phase shifter may include one ormore piezoelectric mirrors. In another implementation, a phase shiftermay include a polaroptic phase shifter that is electrically tunableusing a liquid crystal driver.

A squeezer may include a device with either one optical input and oneoptical output or two optical inputs and two optical outputs. A squeezermay reversibly transform an optical state in a manner that enables ameasurement of one field quadrature of the optical output, with anaccuracy higher than a vacuum noise fluctuation level, at a cost ofbeing able to measure an orthogonal quadrature with a proportionatelylower accuracy. Thus, the squeezer may squeeze a quadrature noisevariance in one direction, while elongating the variance in anotherdirection. A squeezer may be implemented using, for example, a nonlinearperiodically poled lithium niobate (PPNL) or periodically poledpotassium titanyl phosphate (PPKTP) crystal and a strong pump laser.

A Kerr nonlinearity device may include a device with two optical inputsand two optical outputs, which exhibits third-order nonlinear chisusceptibility. A Kerr nonlinearity device may be implemented using anatom, capable of an electromagnetically induced transparency, embeddedinside a photonic crystal microcavity. The embedded atom may impart api-rad-peak phase shift in response to a single photon excitation. AKerr nonlinearity device may implement an effect called a cross-Kerreffect or cross-phase modulation, and may be used for realizing anall-optical two-qubit control-NOT quantum logic gate.

Each of a beam splitter, phase shifter, squeezer, and Kerr nonlinearitydevice may be represented with a corresponding Hamiltonian operator. AnyHamiltonian operator may be decomposed to a linear combination ofcommutation operators. By decomposing the unitary transformationoperator into a linear combination of commutation operators andexpressing the linear combination of commutation operators as acombination of the set of Hamiltonians representing a beam splitter, ashifter, a squeezer, and a Kerr nonlinearity device, the unitarytransformation operator may be expressed as a combination of operatorscorresponding to the set of a beam splitter, a shifter, a squeezer, anda Kerr nonlinearity device.

A layout of a photonic circuit may be generated comprising of beamsplitters, phase shifters, squeezers, and Kerr nonlinearity devices(block 1530). For example, computer device 1200 may generate a layout ofa photonic circuit, which corresponds to the unitary transformationoperator, where the photonic circuit corresponds to a combination ofbeam splitters, phase shifters, squeezers, and Kerr nonlinearitydevices.

The foregoing description provides illustration and description, but isnot intended to be exhaustive or to limit the invention to the preciseform disclosed. Modifications and variations are possible in light ofthe above teachings or may be acquired from practice of the invention.

For example, while series of blocks have been described with respect toFIGS. 11 and 13-15, the order of the blocks may be modified in otherimplementations. Further, non-dependent blocks may be performed inparallel.

Also, certain portions of the implementations may have been described asa “component” that performs one or more functions. The term “component”may include hardware, such as a processor, an ASIC, or a FPGA, or acombination of hardware and software (e.g., software running on aprocessor).

It will be apparent that aspects, as described above, may be implementedin many different forms of software, firmware, and hardware in theimplementations illustrated in the figures. The actual software code orspecialized control hardware used to implement these aspects should notbe construed as limiting. Thus, the operation and behavior of theaspects were described without reference to the specific softwarecode—it being understood that software and control hardware could bedesigned to implement the aspects based on the description herein.

It should be emphasized that the term “comprises/comprising” when usedin this specification is taken to specify the presence of statedfeatures, integers, steps, or components, but does not preclude thepresence or addition of one or more other features, integers, steps,components, or groups thereof.

Even though particular combinations of features are recited in theclaims and/or disclosed in the specification, these combinations are notintended to limit the invention. In fact, many of these features may becombined in ways not specifically recited in the claims and/or disclosedin the specification.

No element, act, or instruction used in the description of the presentapplication should be construed as critical or essential to theinvention unless explicitly described as such. Also, as used herein, thearticle “a” is intended to include one or more items. Where only oneitem is intended, the term “one” or similar language is used. Further,the phrase “based on,” as used herein is intended to mean “based, atleast in part, on” unless explicitly stated otherwise.

1. An optical receiver comprising: a unitary transformation operator to:receive an n-symbol optical codeword associated with a codebook, andperform a unitary transformation on the received optical codeword togenerate a transformed optical codeword, where the unitarytransformation is based on the codebook; n optical detectors, where aparticular one of the n optical detectors is to: detect a particularoptical symbol of the transformed optical codeword, and determinewhether the particular optical symbol corresponds to at least a firstoptical symbol or a second optical symbol; and a decoder to: construct acodeword based on the determinations, and decode the constructedcodeword into a message using the codebook.
 2. The optical receiver ofclaim 1, where the particular one of the n optical detectors includes aDolinar receiver.
 3. The optical receiver of claim 2, where the Dolinarreceiver includes: a first beam splitter to split an incoming beam intoa first beam and a second beam; a second beam splitter to combine thefirst beam and the second beam into a combined beam; a photon detectorto perform a photon count on the combined beam; a processor to select afirst optical symbol or a second optical symbol based on the photoncount; and a feedback controller to control an amplitude and a phase ofthe second beam based on the photon count.
 4. The optical receiver ofclaim 1, where the n-symbol optical codeword corresponds to a two symbolcodeword, where the unitary transformation operator includes a beamsplitter, and where the n optical detectors include a single photondetector and a Dolinar receiver.
 5. The optical receiver of claim 1,where, for an M-ary modulation alphabet, the particular optical symbolcorresponds to a particular symbol selected from three or more opticalsignals.
 6. The optical receiver of claim 1, where the n opticaldetectors perform n projective measurements on the transformed opticalcodeword.
 7. The optical receiver of claim 1, where the n-symbolcodeword is based on a Hadamard matrix, where the unitary transformationoperator includes n*log₂(n/2) beamsplitters, and where the n opticaldetectors include n single photon detectors.
 8. The optical receiver ofclaim 1, where the unitary transformation operator includes a combinedunitary transformation operator that performs a unitary transformationon the n-symbol optical codeword followed by a plurality of single modeunitary transformations on particular optical symbols of the transformedoptical codeword.
 9. The optical receiver of claim 1, where the noptical detectors include n photon number resolving detectors.
 10. Theoptical receiver of claim 1, where the n-symbol optical codewordcorresponds to a temporally encoded codeword, and where the opticalreceiver further includes: an optical demultiplexer to select aparticular optical symbol of the temporally encoded codeword; and noptical buffers, where a particular one of the n optical buffers is toreceive the particular optical symbol, and where the n optical buffersare to forward all optical symbols of the temporally encoded codeword tothe unitary transformation operator at a substantially same time. 11.The optical receiver of claim 1, where the n-symbol codeword is encodedbased on a first order Reed-Muller code, where the unitarytransformation operator includes n*log₂(n/2) beamsplitters, where the noptical detectors include n single photon detectors, where the opticalreceiver further comprises a Dolinar receiver, and where a first one ofthe n single photon detectors to detect a photon is to trigger theDolinar receiver.
 12. A method performed by an optical receiver, themethod comprising: receiving, by the optical receiver, an opticalcodeword associated with a codebook, the optical codeword comprising nsymbols; performing, by the optical receiver, a unitary transformationon the received optical codeword to generate a transformed opticalcodeword comprising n transformed symbols, where the unitarytransformation corresponds to a lossless transformation on the quantumstate of the n symbols; determining, by the optical receiver and for then transformed symbols, whether a particular one of the n transformedsymbols corresponds to at least a first optical symbol or a secondoptical symbol; and generating, by the optical receiver, an electricalcodeword, corresponding to the optical codeword, based on thedetermination.
 13. The method of claim 12, where determining whether theparticular one of the n transformed symbols corresponds to a firstoptical symbol or a second optical symbol is performed by a Dolinarreceiver.
 14. The method of claim 12, where the n-symbol opticalcodeword corresponds to a two symbol codeword, where the unitarytransformation includes a beam splitting operation, and wheredetermining whether the particular one of the n transformed symbolscorresponds to a first optical symbol or a second optical symbol isperformed by a Dolinar receiver in combination with a single photondetector.
 15. The method of claim 12, where, for an M-ary modulationalphabet, the particular optical symbol corresponds to a particularsymbol selected from three or more optical signals.
 16. The method ofclaim 12, where the n-symbol codeword is based on a Hadamard matrix,where the unitary transformation includes n*log₂(n/2) beam splittingoperations, and where determining whether the particular one of the ntransformed symbols corresponds to a first optical symbol or a secondoptical symbol is performed by n single photon detectors.
 17. The methodof claim 12, where performing the unitary transformation on the receivedoptical codeword includes: performing a unitary transformation on thereceived optical codeword; and performing a plurality of single modeunitary transformations on particular optical symbols of the transformedoptical codeword.
 18. The method of claim 12, where the n-symbolcodeword is encoded based on a first order Reed-Muller code, where theunitary transformation includes n*log₂(n/2) beam splitting operations,and where determining whether the particular one of the n transformedsymbols corresponds to a first optical symbol or a second optical symbolis performed by n single photon detectors in combination with a Dolinarreceiver or a Kennedy receiver triggered by a first one of the n singlephoton detectors that detects a photon.
 19. A method performed by acomputer device, the method comprising: receiving, by the computerdevice, a codebook; generating, by the computer device, a minimumprobability of error (MPE) basis for the codebook; generating, by thecomputer device, a measurement unitary matrix for the codebook byexpressing codewords of the codebook in the MPE basis; generating, bythe computer device, a single mode MPE basis based on representations ofsingle mode measurements on binary symbols used to represent thecodewords of the codebook; generating, by the computer device, a singlemode measurement unitary matrix for the codebook by expressing thecodewords of the codebook in the single mode MPE basis; and generating,by the computer device, a unitary transformation operator for thecodebook by multiplying an inverse of the measurement unitary matrixwith the single mode measurement unitary matrix.
 20. The method of claim19, further comprising: decomposing the unitary transformation operatorinto a set of elementary unitary operators; and generating a layout fora photonic circuit based on the decomposed unitary transformationoperator.
 21. The method of claim 20, where the set of elementaryunitary operators includes a beam splitter operator, a phase shifteroperator, a squeezer operator, and a Kerr nonlinearity operator.
 22. Adevice comprising: a joint detection optical receiver that achievessupperadditivite channel capacity by making joint measurements overmultiple optical symbols.
 23. A device comprising: a joint detectionoptical receiver that achieves a Holevo limit, associated with binarymodulation, for an optical channel, when using a codebook associatedwith an optimal code.